Atlas › Modular forms › Weight 10 › Level 15 › 15.10.a

Newspace 15.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(15))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
15.10.a.a	1	$\mathbb{Q}$	$q - 4q^{2} + 81q^{3} - 496q^{4} + 625q^{5} - 324q^{6} - 7680q^{7} + O(q^{8})$	− −	$-1$
15.10.a.b	1	$\mathbb{Q}$	$q + 22q^{2} - 81q^{3} - 28q^{4} - 625q^{5} - 1782q^{6} - 5988q^{7} + O(q^{8})$	+ +	$-1$
15.10.a.c	2	$\mathbb{Q}(\sqrt{4729})$	$2q + 19q^{2} + 162q^{3} + 1521q^{4} - 1250q^{5} + 1539q^{6} - 11872q^{7} + O(q^{8})$	− +	$+1$
15.10.a.d	2	$\mathbb{Q}(\sqrt{241})$	$2q + 31q^{2} - 162q^{3} + 541q^{4} + 1250q^{5} - 2511q^{6} + 14112q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
15.10.a.a	$x + 7684$
15.10.a.b	$x + 5966$
15.10.a.c	$x^{2} + 11853x + 31547096$
15.10.a.d	$x^{2} - 14143x + 22554706$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(15))$	6
Old forms	10
Cusp forms $S_{10}(\Gamma_0(15))$	16
Eisenstein series $E_{10}(\Gamma_0(15))$	4
Modular forms $M_{10}(\Gamma_0(15))$	20

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(15)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 3, 5, 15]: [1, 4, 5, 20] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 3 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(15, 10)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 15:10 1000

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